Stability Derivatives Avl

Stability Derivative Estimation (flight-mechanics/stability-control/stability-derivatives-avl)

Use when the task is estimating the aerodynamic stability derivatives of an aircraft from its geometry, in the style of vortex-lattice (AVL) and DATCOM-type preliminary methods: wing and tail lift curve slopes, pitch stiffness, directional and roll stability derivatives, and the neutral point with the static margin.

This leaf ESTIMATES the derivatives from geometry. It is different from longitudinal-stability (which computes the neutral point from given coefficients) and from dynamic-stability (which classifies modes from given derivatives): start here when the input is the wing and tail planform, then hand the derivative table to those leaves.

Domain quick reference

Conventions: stability axes, x forward, y out the right wing, z down. All angles in radians, all lengths in meters, all areas in square meters. Every derivative below is non-dimensional, per radian of the relevant motion variable, unless stated otherwise. The roll and yaw rate variables are the non-dimensional rates pb/2V and rb/2V.

• Wing lift curve slope, per radian (subsonic planform estimate, valid for Mach below 0.9):

  CL_alpha = 2*pi*A / (2 + sqrt( (A*beta/k)^2 * (1 + tan^2(Lambda)/
             beta^2) + 4 ))
  

  with A the aspect ratio (dimensionless), Lambda the quarter-chord sweep in radians, beta = sqrt(1 - M^2) the compressibility factor (dimensionless), and k = a0 / (2pi) the section slope ratio, where a0 is the airfoil section lift slope, 2pi per radian by default. The tail lift curve slope uses the same formula with the tail surface aspect ratio and sweep.

• Downwash gradient at the tail, dimensionless (elliptical-loading estimate):

  depsilon/dalpha = 2 * CL_alpha_w / (pi * A)
  

• Horizontal tail volume coefficient, dimensionless (tail arm l_t in m, tail area S_t in m^2, mean aerodynamic chord c_bar in m, wing area S_w in m^2):

  V_h = l_t * S_t / (c_bar * S_w)
  

• Vertical tail volume coefficient, dimensionless (vertical tail arm l_v in m, vertical tail area S_v in m^2, span b in m):

  V_v = l_v * S_v / (b * S_w)
  

• Pitch stiffness Cm_alpha, per radian, referenced to the center of gravity h_cg (positions are fractions of the mean aerodynamic chord):

  Cm_alpha = a_w * (h_cg - h_ac_w) - V_h * a_t * (1 - depsilon/dalpha)
  

  with h_ac_w the wing aerodynamic center and a_w, a_t the wing and tail lift slopes. Negative Cm_alpha means pitch stable.

• Neutral point, fraction of mean aerodynamic chord:

  h_np = h_ac_w + V_h * (a_t / a_w) * (1 - depsilon/dalpha)
  

• Static margin = h_np - h_cg, dimensionless; positive means a stable configuration.

• Directional stability Cn_beta, per radian: the vertical tail contribution is stabilizing (positive weathercock term), the wing-body sweep contribution is destabilizing (negative):

  Cn_beta = V_v * a_t * (1 + sigma) - (C_L^2 / (pi * A)) * tan(Lambda)
  

  with sigma the sidewash factor (0.72 common preliminary value, dimensionless) and C_L the cruise lift coefficient.

• Dihedral contribution to roll stability Cl_beta, per radian:

  Cl_beta = -(a_w / 2) * Gamma
  

  with Gamma the wing dihedral angle in radians. Negative means stabilizing in sideslip.

• Roll damping Cl_p and yaw damping Cn_r, per radian of pb/2V and rb/2V (lambda is the taper ratio, dimensionless, in (0, 1]):

  Cl_p = -(a_w / 12) * (1 + 3*lambda) / (1 + lambda)
  Cn_r = -2 * V_v * a_t * (l_v / b)
  

• Cross derivatives Cl_r and Cn_p, per radian (wing estimates):

  Cl_r = (C_L / 4) * (1 + 3*lambda) / (1 + lambda)
  Cn_p = -C_L / 8
  

Workflow

1. Collect the wing planform: aspect ratio, quarter-chord sweep (degrees in, converted to radians inside), taper ratio, airfoil section slope, and the Mach number and cruise lift coefficient.
2. Estimate the wing and tail lift curve slopes with cl_alpha_wing and cl_alpha_tail, then the downwash gradient with downwash_gradient.
3. Build the tail volume coefficients with tail_volume_coeff and vertical_tail_volume_coeff from the tail arms, areas, mean chord, span, and wing area.
4. Estimate Cm_alpha with cm_alpha; derive the neutral point with neutral_point and the static margin with static_margin.
5. Estimate the lateral-directional derivatives with cn_beta, cl_beta, cl_p, cl_r, cn_p, and cn_r from the vertical tail, the dihedral, and the sweep.
6. Assemble the full non-dimensional table with estimate_derivative_table and read the verdicts: pitch_stable, directionally_stable, and statically_stable.
7. Hand the derivative table to the longitudinal-stability and dynamic-stability leaves for the neutral-point check and the mode classification.

Pitfalls

• Mixing units: the formulas take sweep and dihedral in degrees and convert internally, but lengths and areas must be SI (meters, square meters); feeding feet or inches misstates every result.
• Mach limit: the subsonic planform formula is valid for M below 0.9; feeding transonic or supersonic Mach numbers raises ValueError instead of returning a misleading slope.
• Reversing the Cm_alpha sign: negative Cm_alpha is pitch stable, positive is unstable; the same sign rule applies to Cl_p and Cn_r (negative is damping) and to Cn_beta (positive is weathercock stable).
• Confusing the neutral point with the wing aerodynamic center: the neutral point includes the tail contribution, the wing aerodynamic center alone does not.
• Downwash gradient at or above 1.0: physically invalid for this model and rejected.
• Treating the estimates as final values: these are preliminary design approximations; validate the final configuration with higher-fidelity tools.
• Zero vertical tail volume: a configuration with no vertical tail cannot be directionally stable under this model; the wing-body contribution is destabilizing.

Behavior contract (gate 3)

The derivative estimation logic is exercised by the gate 3 contract test: scripts/test_stability_derivatives_avl.py against scripts/stability_derivatives_avl_logic.py (stdlib unittest, offline). Run:

python3 scripts/test_stability_derivatives_avl.py

Compliance

• Standards referenced, not reproduced: FAR-25 and CS-25 require positive static longitudinal stability and adequate directional stability for transport aeroplanes; the derivative estimation formulas are common flight mechanics methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.